Hyperboloid of One Sheet

Differential Equations, Mechanics, and Computation
hyperboloid
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The hyperboloid Of One Sheet is a surface of revolution of the curve family hyperbola.

The parametric formula for the Hyperboloid of One Sheet is:

ParametricPlot3D[{Cosh[u]*Cos[v], Cosh[u]*Sin[v], Sinh[u]}, {u, -2, 2}, {v, 0, 2*π}]
(* u → height, v → circular sweep. *)

hyperboloid1.nb.zip

Ruled Surface

hyperboloid of one sheet
A Hyperboloid of one sheet, showing its ruled surface property.

A revolving around its transverse axis forms a surface called “hyperboloid of one sheet”. A hyperboloid is a Ruled Surface.

Ruled surfaces are surfaces that for every point on the surface, there is a line on the surface passing it. Or, in other words, a surface generated by a line. If for each point on the surface there are two lines on the surface passing it, then it's called doubly-ruled surface. Hyperboloid is a doubly-ruled surface.

Ruled surfaces also include cylinder and helicoid. There are only 3 doubly ruled surfaces: The hyperboloid, hyperbolic paraboloid, and plane.

hyperboloid sculpture
Two hyperboloid sculptures. The left can be made into a modern fruit basket, and the right a paper waste basket. (source: Second Life gallery)
hyperbolic basket
Photo of a real basket modeled by hyperboloid.

Spinning Cube Silhouette

The silhouette of a rotating dice is a hyperbola.

spinning cube
A sculpture emulating a spinning cube. (source: Second Life gallery)
spinning dice photo
A photo of a spinning dice. (See: Photo of regular polyhedron dices)

Hyperboloid in Architecture

Due to its simplicity and beauty, the hyperboloid is often used in architecture for towers. They are called Hyperboloid structure.

James S McDonnell Planetarium
The James S McDonnell Planetarium building at the St. Louis Science Center.
Kobe port tower
The Kobe Port Tower. 2142×3200 🌐 img src

For more photos, see: Hyperboloid Towers.

hypoid
Hyperboloid can be used for gear surface, called a Hypoid. (Source: Penrose Tiles to Trapdoor Ciphers by Martin Gardner. amazon, chapter 15).

Other algebaric surfaces that has cross-sections of conic sections are: ellipsoid, paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets.

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